OS: Difference between revisions
Cmloegcmluin (talk | contribs) break up wall of information into helpful sections that are consistent across all arithmetic tuning pages |
Cmloegcmluin (talk | contribs) |
||
| (7 intermediate revisions by 3 users not shown) | |||
| Line 3: | Line 3: | ||
== Specification == | == Specification == | ||
Its full specification is (n-)OSp: (n pitches of an) otonal sequence adding by rational interval p. The "n" is optional. If unspecified, you describe an open-ended sequence. | Its full specification is (n-)OSp: (n pitches of an) [[otonal]] sequence adding by rational interval p. The "n" is optional. If unspecified, you describe an open-ended sequence. | ||
== Formula == | == Formula == | ||
| Line 23: | Line 23: | ||
== Relationship to other tunings == | == Relationship to other tunings == | ||
=== | === Vs. AFS === | ||
An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n- | An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-OSp must be rational. | ||
=== As shifted overtone series === | === As shifted overtone series === | ||
Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is irrational). By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation|derivation]]. | Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is probably irrational). By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation|derivation]]. | ||
=== | Yet another term for this structure is an [[isoharmonic series]]. | ||
=== Vs. OD === | |||
By specifying n, your OS will be equivalent to some [[OD|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7. | By specifying n, your OS will be equivalent to some [[OD|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7. | ||
=== | === Vs. US === | ||
The analogous undertone equivalent of an OS is a [[US]]. | The analogous undertone equivalent of an OS is a [[US]]. | ||
| Line 55: | Line 57: | ||
! 8 | ! 8 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
|(4/4) | |(4/4) | ||
|7/4 | |7/4 | ||
| Line 66: | Line 68: | ||
|28/4 | |28/4 | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
|(0) | |(0) | ||
|0.81 | |0.81 | ||
| Line 77: | Line 79: | ||
|2.81 | |2.81 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
|(1/1) | |(1/1) | ||
|4/7 | |4/7 | ||
| Line 137: | Line 139: | ||
So we can see that <span><math>\frac 13</math></span> was the right amount to shift by because it is the delta from the starting position <span><math>1</math></span> to <span><math>\frac 43</math></span>, the latter of which is the reciprocal of the target step size <span><math>\frac 34</math></span> and therefore the value that we need the starting position to equal in order to be sent ''back'' to <span><math>1</math></span> when we resize all steps from 1 to the target step size by multiplying everything by it. | So we can see that <span><math>\frac 13</math></span> was the right amount to shift by because it is the delta from the starting position <span><math>1</math></span> to <span><math>\frac 43</math></span>, the latter of which is the reciprocal of the target step size <span><math>\frac 34</math></span> and therefore the value that we need the starting position to equal in order to be sent ''back'' to <span><math>1</math></span> when we resize all steps from 1 to the target step size by multiplying everything by it. | ||
[[Category:Otonality]] | [[Category:Otonality]] | ||
[[Category:Harmonic]] | [[Category:Harmonic]] | ||
[[Category:Harmonic series]] | [[Category:Harmonic series]] | ||
[[Category:Xenharmonic series]] | |||